Theorem soisores 6577

Description: Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)

Assertion

Ref	Expression
soisores	⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem soisores

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isorel 6576	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦)))
2		fvres 6207	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑥) = (𝐹‘𝑥))
3		fvres 6207	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
4	2, 3	breqan12d 4669	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
5	4	adantl 482	. . . . 5 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑥)𝑆((𝐹 ↾ 𝐴)‘𝑦) ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
6	1, 5	bitrd 268	. . . 4 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ↔ (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
7	6	biimpd 219	. . 3 ⊢ (((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
8	7	ralrimivva 2971	. 2 ⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
9		ffn 6045	. . . . . . . 8 ⊢ (𝐹:𝐵⟶𝐶 → 𝐹 Fn 𝐵)
10	9	ad2antrl 764	. . . . . . 7 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → 𝐹 Fn 𝐵)
11		simprr 796	. . . . . . 7 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → 𝐴 ⊆ 𝐵)
12		fnssres 6004	. . . . . . 7 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝐹 ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	syl2anc 693	. . . . . 6 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → (𝐹 ↾ 𝐴) Fn 𝐴)
14	13	3adant3 1081	. . . . 5 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝐹 ↾ 𝐴) Fn 𝐴)
15		df-ima 5127	. . . . . . 7 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴)
16	15	eqcomi 2631	. . . . . 6 ⊢ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴)
17	16	a1i 11	. . . . 5 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴))
18		fvres 6207	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑧) = (𝐹‘𝑧))
19		fvres 6207	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝑤) = (𝐹‘𝑤))
20	18, 19	eqeqan12d 2638	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
21	20	adantl 482	. . . . . . 7 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧) = (𝐹‘𝑤)))
22		simprl 794	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑧 ∈ 𝐴)
23		simprr 796	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐴)
24		simpl3 1066	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))
25		breq1 4656	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑧𝑅𝑦))
26		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
27	26	breq1d 4663	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑦)))
28	25, 27	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝑧𝑅𝑦 → (𝐹‘𝑧)𝑆(𝐹‘𝑦))))
29		breq2 4657	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑤))
30		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
31	30	breq2d 4665	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑧)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
32	29, 31	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → ((𝑧𝑅𝑦 → (𝐹‘𝑧)𝑆(𝐹‘𝑦)) ↔ (𝑧𝑅𝑤 → (𝐹‘𝑧)𝑆(𝐹‘𝑤))))
33	28, 32	rspc2va 3323	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝑧𝑅𝑤 → (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
34	22, 23, 24, 33	syl21anc 1325	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 → (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
35		breq1 4656	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝑥𝑅𝑦 ↔ 𝑤𝑅𝑦))
36		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
37	36	breq1d 4663	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑥)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑦)))
38	35, 37	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → ((𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) ↔ (𝑤𝑅𝑦 → (𝐹‘𝑤)𝑆(𝐹‘𝑦))))
39		breq2 4657	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑤𝑅𝑦 ↔ 𝑤𝑅𝑧))
40		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
41	40	breq2d 4665	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑤)𝑆(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑆(𝐹‘𝑧)))
42	39, 41	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑤𝑅𝑦 → (𝐹‘𝑤)𝑆(𝐹‘𝑦)) ↔ (𝑤𝑅𝑧 → (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
43	38, 42	rspc2va 3323	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝑤𝑅𝑧 → (𝐹‘𝑤)𝑆(𝐹‘𝑧)))
44	23, 22, 24, 43	syl21anc 1325	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑤𝑅𝑧 → (𝐹‘𝑤)𝑆(𝐹‘𝑧)))
45	34, 44	orim12d 883	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
46	45	con3d 148	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧)) → ¬ (𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧)))
47		simpl1r 1113	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑆 Or 𝐶)
48		simpl2l 1114	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝐹:𝐵⟶𝐶)
49		simpl2r 1115	. . . . . . . . . . 11 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝐴 ⊆ 𝐵)
50	49, 22	sseldd 3604	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑧 ∈ 𝐵)
51	48, 50	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘𝑧) ∈ 𝐶)
52	49, 23	sseldd 3604	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑤 ∈ 𝐵)
53	48, 52	ffvelrnd 6360	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐶)
54		sotrieq 5062	. . . . . . . . 9 ⊢ ((𝑆 Or 𝐶 ∧ ((𝐹‘𝑧) ∈ 𝐶 ∧ (𝐹‘𝑤) ∈ 𝐶)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
55	47, 51, 53, 54	syl12anc 1324	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
56		simpl1l 1112	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → 𝑅 Or 𝐵)
57		sotrieq 5062	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧 = 𝑤 ↔ ¬ (𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧)))
58	56, 50, 52, 57	syl12anc 1324	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 ↔ ¬ (𝑧𝑅𝑤 ∨ 𝑤𝑅𝑧)))
59	46, 55, 58	3imtr4d 283	. . . . . . 7 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧) = (𝐹‘𝑤) → 𝑧 = 𝑤))
60	21, 59	sylbid 230	. . . . . 6 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) → 𝑧 = 𝑤))
61	60	ralrimivva 2971	. . . . 5 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) → 𝑧 = 𝑤))
62		dff1o6 6531	. . . . 5 ⊢ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴) Fn 𝐴 ∧ ran (𝐹 ↾ 𝐴) = (𝐹 “ 𝐴) ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝐹 ↾ 𝐴)‘𝑧) = ((𝐹 ↾ 𝐴)‘𝑤) → 𝑧 = 𝑤)))
63	14, 17, 61, 62	syl3anbrc 1246	. . . 4 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴))
64		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
65	64	a1i 11	. . . . . . . . . 10 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤)))
66	65, 44	orim12d 883	. . . . . . . . 9 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑧 = 𝑤 ∨ 𝑤𝑅𝑧) → ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
67	66	con3d 148	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (¬ ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧)) → ¬ (𝑧 = 𝑤 ∨ 𝑤𝑅𝑧)))
68		sotric 5061	. . . . . . . . 9 ⊢ ((𝑆 Or 𝐶 ∧ ((𝐹‘𝑧) ∈ 𝐶 ∧ (𝐹‘𝑤) ∈ 𝐶)) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
69	47, 51, 53, 68	syl12anc 1324	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) ↔ ¬ ((𝐹‘𝑧) = (𝐹‘𝑤) ∨ (𝐹‘𝑤)𝑆(𝐹‘𝑧))))
70		sotric 5061	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐵 ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧𝑅𝑤 ↔ ¬ (𝑧 = 𝑤 ∨ 𝑤𝑅𝑧)))
71	56, 50, 52, 70	syl12anc 1324	. . . . . . . 8 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 ↔ ¬ (𝑧 = 𝑤 ∨ 𝑤𝑅𝑧)))
72	67, 69, 71	3imtr4d 283	. . . . . . 7 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝐹‘𝑧)𝑆(𝐹‘𝑤) → 𝑧𝑅𝑤))
73	34, 72	impbid 202	. . . . . 6 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
74	18, 19	breqan12d 4669	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
75	74	adantl 482	. . . . . 6 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤) ↔ (𝐹‘𝑧)𝑆(𝐹‘𝑤)))
76	73, 75	bitr4d 271	. . . . 5 ⊢ ((((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝑧𝑅𝑤 ↔ ((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤)))
77	76	ralrimivva 2971	. . . 4 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ ((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤)))
78		df-isom 5897	. . . 4 ⊢ ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ((𝐹 ↾ 𝐴):𝐴–1-1-onto→(𝐹 “ 𝐴) ∧ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 ↔ ((𝐹 ↾ 𝐴)‘𝑧)𝑆((𝐹 ↾ 𝐴)‘𝑤))))
79	63, 77, 78	sylanbrc 698	. . 3 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))) → (𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)))
80	79	3expia 1267	. 2 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)) → (𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴))))
81	8, 80	impbid2 216	1 ⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574   class class class wbr 4653   Or wor 5034  ran crn 5115   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  isercolllem1  14395  dvgt0lem2  23766  erdszelem4  31176  erdszelem8  31180  erdsze2lem2  31186